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Spherical Coordinates Brick Demonstration

Spherical Coordinates Brick Demonstration
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2017-02-21 07:32
TrigonometryCoordinate GeometrySolid GeometryCalculusParametric Equation
This example demonstrates how to plot a brick shape in spherical coordinates. It involves core math concepts like spherical coordinates, region representation, and 3D geometry. Construction highlights include commands like PointIn, If, CurveCartesian, Point, Surface, Intersect, Segment, CircumcircleArc, and CircumcircleSector. Suitable for high school and above students learning about applications of spherical coordinates.

GeoGebra Commands & Parameters

  1. PointIn(o) → A
  2. If(x(A) > 0, atan(y(A) / x(A)), atan(y(A) / x(A)) + π) → α_1
  3. CurveCartesian((cos(α_1 + t) * sin(β_1)), (sin(α_1 + t) * sin(β_1)), cos(β_1), t, 0, 3.141592653589793) → c
  4. Point(c) → B
  5. If(x(B) > 0, atan(y(B) / x(B)), atan(y(B) / x(B)) + π) → α_2
  6. CurveCartesian((cos(α_1) * sin(β_1 + t)), (sin(α_1) * sin(β_1 + t)), cos(β_1 + t), t, 0, π - β_1) → d
  7. Point(d) → C
  8. CurveCartesian((cos(α_1 + t) * sin(β_1)), (sin(α_1 + t) * sin(β_1)), cos(β_1), t, 0, Δα) → e
  9. Surface((sin(u) * cos(t)), (sin(u) * sin(t)), cos(u), u, β_1, β_2, t, α_1, α_2) → a
  10. Intersect(yAxis, zAxis) → E
  11. Segment(A, E) → f
  12. Point(f) → F
  13. Segment(E, F) → h
  14. Segment(F, A) → i
  15. Surface(((h * sin(u)) * cos(t)), ((h * sin(u)) * sin(t)), (h * cos(u)), u, β_1, β_2, t, α_1, α_2) → b
  16. Surface(((t * sin(β_1)) * cos(u)), ((t * sin(β_1)) * sin(u)), (t * cos(β_1)), u, α_1, α_2, t, h, 1) → j
  17. Surface(((t * sin(β_2)) * cos(u)), ((t * sin(β_2)) * sin(u)), (t * cos(β_2)), u, α_1, α_2, t, h, 1) → k
  18. Surface(((t * sin(u)) * cos(α_1)), ((t * sin(u)) * sin(α_1)), (t * cos(u)), u, β_1, β_2, t, h, 1) → l
  19. Surface(((t * sin(u)) * cos(α_2)), ((t * sin(u)) * sin(α_2)), (t * cos(u)), u, β_1, β_2, t, h, 1) → m
  20. CurveCartesian((cos(α_1 + t) * sin(β_2)), (sin(α_1 + t) * sin(β_2)), cos(β_2), t, 0, Δα) → g
  21. CurveCartesian((cos(α_1) * sin(β_1 + t)), (sin(α_1) * sin(β_1 + t)), cos(β_1 + t), t, 0, Δβ) → n
  22. CurveCartesian((cos(α_2) * sin(β_1 + t)), (sin(α_2) * sin(β_1 + t)), cos(β_1 + t), t, 0, Δβ) → p
  23. Segment(B, B') → q
  24. Segment(C, C') → r
  25. Segment(D, D') → s
  26. CurveCartesian(((h * cos(α_1 + t)) * sin(β_1)), ((h * sin(α_1 + t)) * sin(β_1)), (h * cos(β_1)), t, 0, Δα) → t
  27. CurveCartesian(((h * cos(α_1 + t)) * sin(β_2)), ((h * sin(α_1 + t)) * sin(β_2)), (h * cos(β_2)), t, 0, Δα) → u
  28. CurveCartesian(((h * cos(α_1)) * sin(β_1 + t)), ((h * sin(α_1)) * sin(β_1 + t)), (h * cos(β_1 + t)), t, 0, Δβ) → v
  29. CurveCartesian(((h * cos(α_2)) * sin(β_1 + t)), ((h * sin(α_2)) * sin(β_1 + t)), (h * cos(β_1 + t)), t, 0, Δβ) → w
  30. CurveCartesian((cos(α_1 + t) * sin(β_1)), (sin(α_1 + t) * sin(β_1)), cos(β_1), t, 0, 6.283185307179586) → a_1
  31. CurveCartesian((cos(α_1 + t) * sin(β_2)), (sin(α_1 + t) * sin(β_2)), cos(β_2), t, 0, 6.28319) → b_1
  32. CurveCartesian((cos(α_1) * sin(t)), (sin(α_1) * sin(t)), cos(t), t, 0, 3.141592653589793) → c_1
  33. CurveCartesian((cos(α_2) * sin(t)), (sin(α_2) * sin(t)), cos(t), t, 0, 3.141592653589793) → d_1
  34. Segment(B, E) → f_1
  35. Segment(C, E) → g_1
  36. Segment(D, E) → h_1
  37. CircumcircleArc(H, C, G) → k_1
  38. CircumcircleArc(H, B, G) → p_1
  39. Segment(I, E) → i_1
  40. Segment(J, E) → j_1
  41. Surface((sin(u) * cos(t)), (sin(u) * sin(t)), cos(u), u, 0, β_1, t, α_1, α_2) → m_1
  42. Surface(((t * sin(u)) * cos(α_2)), ((t * sin(u)) * sin(α_2)), (t * cos(u)), u, β_1, β_2, t, 0, h) → n_1
  43. Surface(((t * sin(u)) * cos(α_2)), ((t * sin(u)) * sin(α_2)), (t * cos(u)), u, 0, β_1, t, 0, 1) → o_1
  44. Surface(((t * sin(u)) * cos(α_2)), ((t * sin(u)) * sin(α_2)), (t * cos(u)), u, β_2, 1.5707963267948966, t, 0, 1) → q_1
  45. Surface(((t * sin(u)) * cos(α_1)), ((t * sin(u)) * sin(α_1)), (t * cos(u)), u, 0, β_1, t, 0, 1) → r_1
  46. Surface(((t * sin(u)) * cos(α_1)), ((t * sin(u)) * sin(α_1)), (t * cos(u)), u, β_2, 1.5708, t, 0, 1) → s_1
  47. Surface(((t * sin(u)) * cos(α_1)), ((t * sin(u)) * sin(α_1)), (t * cos(u)), u, β_1, β_2, t, 0, h) → t_1
  48. CurveCartesian(cos(t), sin(t), 0, t, 0, 1.5707963267948966) → e_1
  49. Surface((sin(u) * cos(t)), (sin(u) * sin(t)), cos(u), u, β_2, 3.141592653589793, t, α_1, α_2) → l_1
  50. Surface((t * cos(α_1)), (t * sin(α_1)), u, t, 0, 1, u, -1, 1) → u_1
  51. Surface((t * cos(α_2)), (t * sin(α_2)), u, t, 0, 1, u, -1, 1) → v_1
  52. CircumcircleSector(I, K, J) → c_2